PX157 - E4a - LRC circuits 1

step 1:

{\tilde{Z}}_{e q} = {\tilde{Z}}_{R} + {\tilde{Z}}_{C} = R + \frac{1}{j ω C}

step 2:

{\tilde{V}}_{i n} = {\tilde{Z}}_{e q} \tilde{I} ⟹ \tilde{I} = \frac{{\tilde{V}}_{i n}}{R + \frac{1}{j ω C}}

step 3:

{\tilde{V}}_{o u t} = {\tilde{Z}}_{C} \tilde{I} = \frac{1}{j ω C} \frac{{\tilde{V}}_{i n}}{R + \frac{1}{j ω C}} = \frac{{\tilde{V}}_{i n}}{j ω R C + 1}

step 4:

| {\tilde{V}}_{o u t} | = \frac{| {\tilde{V}}_{i n} |}{\sqrt{ω^{2} R^{2} C^{2} + 1}} ⟹ \frac{| {\tilde{V}}_{o u t} |}{1} = \frac{| {\tilde{V}}_{i n} |}{\sqrt{ω^{2} R^{2} C^{2} + 1}}

circuit acts as a low-pass filter
typical relaxation time: $τ = R C$
step 5:

\begin{aligned} {\tilde{Z}}_{e q} & = Z_{0} e^{j ϕ} \\ ϕ & = \arctan (\frac{I m {\tilde{Z}}_{e q}}{Re {\tilde{Z}}_{e q}}) \\ = \arctan (- \frac{1}{ω R C}) \end{aligned}

${\tilde{V}}_{i n} = {\tilde{Z}}_{e q} \tilde{I} = Z_{0} e^{i ϕ} \tilde{I}$ $\tilde{V}$ leads $\tilde{I}$ by a phase, $ϕ$
since $ϕ < 0$ , ${\tilde{V}}_{i n}$ lags behind $\tilde{I}$